Timeline for Vietoris-Rips complex homology of a higher degree than the ambient dimension

7 events

when toggle format	what		by	license	comment
Apr 13, 2019 at 14:47	comment	added	Mustafa		@AdamP.Goucher I would conjecture that your example with $2n+2$ points is the only one that yields $S^n$.
Apr 13, 2019 at 12:07	comment	added	Adam P. Goucher		@Mustafa I can't immediately find any larger examples which are homeomorphic to $S^n$. The main obstruction is that once you have $n + 1$ vertices forming an $n$-simplex, you're not allowed to add any other vertices within its convex hull. But if you want $S^n$, then any point in general position in $\mathbb{R}^2$ must lie within the convex hull of an even number of $n$-simplices.
Apr 13, 2019 at 3:50	comment	added	Mustafa		Nice example! Is there any other example of finite sets $X\subset \mathbb{R}^2$ (equipped with the $l_2$ metric) that yields $S^n$ as its Vietoris-Rips complex for some $n$?
Jun 18, 2018 at 16:59	comment	added	j.c.		Very nice - I added an image from Wikipedia.
Jun 18, 2018 at 16:59	history	edited	j.c.	CC BY-SA 4.0	add image and link
Jun 18, 2018 at 14:50	vote	accept	Arkadi
Jun 18, 2018 at 8:39	history	answered	Adam P. Goucher	CC BY-SA 4.0